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Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor.

1 Introduction

Let [phi]: [M.sup.n] [right arrow] [[bar.M].sup.n] be an isometric immersion from an n-dimensional Riemannian manifold into a complex n-dimensional Kahler manifold [[bar.M].sup.n]. [M.sup.n] is called a Lagrangian submanifold if the almost complex structure J of [[bar.M].sup.n] carries each tangent space of [M.sup.n] into its corresponding normal space.

In this paper, we study Lagrangian submanifolds of complex space forms [[bar.M].sup.n] (4c) with constant holomorphic sectional curvature 4c. In particular we are interested in Lagrangian submanifolds of

(i) [[bar.M].sup.n] (4c) = [[??].sup.n], when c = 0,

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the basic existence and uniqueness theorem it follows that such Lagrangian submanifolds are completely determined by the metric and the (totally symmetric) cubic form (h(X, Y), JZ). Here h denotes the second fundamental form of the immersion. In that respect, it is natural to look for Lagrangian submanifolds for which this cubic form (or the underlying second fundamental form) satisfies some geometric properties. One of the most natural properties in this viewpoint is the notion of isotropic submanifold introduced by O'Neill ([15]). A Lagrangian submanifold is called isotropic if and only if there exists a constant [LAMBDA](p) such that for every unit vector v [member of] [T.sub.p][M.sup.n], [parallel]h(v, v)[parallel] = [LAMBDA](p). If moreover A is independent of the point p, then Mn is called constant isotropic. Note that a 2-dimensional minimal Lagrangian surface is always isotropic. In higher dimensions, isotropic Lagrangian submanifolds were studied and completely classified in [10], [13], [14] and [18]. Studying these classifications it follows immediately that such submanifolds necessarily need to have parallel second fundamental form or are H-umbilical in the sense of [7] and [8].

Here in this paper, we study Lagrangian submanifolds [M.sup.n] in complex space forms with isotropic cubic tensor, i.e. there exists a real function Y on [M.sup.n] such that for any unit tangent vector v at a point p we have that

[parallel]([nabla]h)(v, v, v)[parallel] = Y( p).

Here the cubic tensor means the derivative of the second fundamental form, i.e., ([nabla]h) (X, Y, Z), which is different from the cubic form <h(X, Y), JZ>. Note that all parallel Lagrangian submanifolds provide trivial examples of such submanifolds.

In [17], L. Su showed that Lagrangian surfaces in 2-dimensional complex space forms with isotropic cubic tensor either have parallel mean curvature vector or are congruent to one of the Whitney spheres (or their analogs in complex hyperbolic space).

In this paper, we deal with the higher dimensional case and give a complete classification of the Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor. We will give the explicit constructions of all such examples in section 3. Our classification theorem implies in particular that some of the Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor are also isotropic submanifolds (in the sense of O'Neill). We also prove the converse, namely that any n-dimensional isotropic Lagrangian submanifold in complex space forms with n [greater than or equal to] 3 has isotropic cubic tensor. More precisely, we show the following results:

Theorem 1.1. Let [M.sup.3] be a Lagrangian submanifold with isotropic cubic tensor of a complex space form. Assume that [M.sup.3] is nowhere parallel, then either

(i) [M.sup.3] is congruent with a Whitney sphere in [[??].sup.3] (see (3.4)) and [??][P.sup.3] (see (3.5)), or their analogs in [??][H.sup.3] (see (3.6), (3.7) and (3.8)), or

(ii) [M.sup.3] is congruent with one of the Examples 6-10 with n = 3 defined in section 3 (see (3.10)-(3.14)).

Theorem 1.2. The Whitney sphere in [[??].sup.n] (see (3.4)) and [??][P.sup.n] (see (3.5)), and their analogs in [??][H.sup.n] (see (3.6), (3.7) and (3.8)) are Lagrangian immersions with isotropic cubic tensor.

Theorem 1.3. Any n-dimensional (n [greater than or equal to] 3) H-umbilical isotropic Lagrangian submanifold of a complex space form has isotropic cubic tensor.

The paper is organized as follows. In Section 2 we will recall the basic formulas for Lagrangian submanifolds of complex space forms. In Section 3 we will recall the construction of some examples, which are fundamental for our paper. We will show in arbitrary dimensions that all these examples have isotropic cubic tensor. In the final two sections we will deal with the converse problem. In Section 4 we will determine all possible isotropic cubic tensors in dimension 3. Finally, in Section 5, in dimension 3, we complete the proof of Theorem 1.1.

2 Preliminaries

In this section, [M.sup.n] will always denote an n-dimensional Lagrangian submanifold of [[bar.M].sup.n] (4c) which is an n-dimensional complex space form with constant holomorphic sectional curvature 4c. We denote the Levi-Civita connections on [M.sup.n], [[bar.M].sup.n] (4c) and the normal bundle by [nabla], D and [[nabla].sup.[perpendicular to].sub.X] respectively. The formulas of Gauss and Weingarten are given by (see [2], [3], [4], [5], [6])

[D.sub.X]Y = [[nabla].sub.X]Y + h(X, Y), [D.sub.X[xi]] = -[A.sub.[xi]]X + [[nabla].sup.[perpendicular to].sub.X][xi], (2.1)

where X and Y are tangent vector fields and C is a normal vector field on [M.sup.n]. The Lagrangian condition implies that (see [6], [7], [8], [18])

[[nabla].sup.[perpendicular to].sub.X]JY = J[[nabla].sub.X]Y, [A.sub.JX]Y = - Jh(X, Y) = [A.sub.JY]X, (2.2)

where h is the second fundamental form and A denotes the shape operator.

We denote the curvature tensors of [nabla] and by [[nabla].sup.[perpendicular to].sub.X] and [R.sup.[perpendicular] to] respectively. The first and second covariant derivatives of h are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where X, Y, Z and W are tangent vector fields.

The equations of Gauss, Codazzi and Ricci for a Lagrangian submanifold of [[bar.M].sup.n] (4c) are given by (see [3], [4], [6], [9])

<R(X, Y)Z,W> = <h(Y,Z),h(X, W)> - <h(X,Z),h(Y, W)>

+ c (<X, W> <Y,Z> - <X,Z> <Y, W>), (2.4)

([nabla]h) (X, Y, Z) = ([nabla]h) ( Y, X, Z), (2.5)

<[R.sup.[perpendicular to]] (X, Y) JZ, JW> = <[[A.sub.JZ] , [A.sub.JW]]X, Y> + c (<X, W> <Y,Z>- <X,Z> <Y, W>),

where X, Y Z and W are tangent vector fields. Note that for a Lagrangian submanifold the equations of Gauss and Ricci are mutually equivalent

We have the following Ricci identity (see [14]):

([[nabla].sup.2]h)(X, Y, Z, W) = ([[nabla].sup.2]h)(Y, X, Z, W) + JR(X, Y) [A.sub.JZ]W - h(R(X, Y)Z, W) - h(R(X, Y)W, Z), (2.7)

where X, Y, Z and W are tangent vector fields. The Lagrangian condition implies that

<[R.sup.[perpendicular to]](X, Y) JZ, JW> = <R(X, Y)Z, W>, (2.8)

<h(X, Y), JZ> = <h(X, Z), JY>, (2.9)

for tangent vector fields X, Y, Z and W. From (2.3) and (2.9), we also have

<([nabla]h)(W, X, Y), JZ> = <([nabla]h)(W, X, Z), JY>, (2.10)

for tangent vector fields X, Y, Z and W.

From now on, we will also assume that [M.sup.n] has an isotropic cubic tensor, i.e. in each point p of [M.sup.n,] [parallel][nabla]h(v, v, v)[parallel] is independent of the unit vector v [member of] [T.sub.p][M.sup.n]. Hence, we obtain a function Y on Mn by

Y(p) = [parallel][nabla]h(v, v, v)[parallel], (2.11)

where v [member of] [T.sub.p][M.sup.n] with [parallel]v[parallel] = 1.

We note that [M.sup.n] is called an isotropic submanifold if at each point p of [M.sup.n], [parallel]h(v, v)[parallel] is independent of the unit vector v [member of] [T.sub.p][M.sup.n] (see [10], [13], [15] and [18]).

3 Basic examples and the proof of Theorem 1.2,1.3

In this section we will recall some basic examples of Lagrangian submanifolds in complex space forms. All of these examples are H-umbilical. Following [7] and [8] a Lagrangian submanifold is called H-umbilical if and only if there exists a local orthonormal frame {[E.sub.1],..., [E.sub.n]} and differentiable functions [lambda] and [mu] such that

h([E.sub.1], [E.sub.1]) = [lambda]J[E.sub.1], h([E.sub.1], [E.sub.1]) = [mu][E.sub.i], h([E.sub.i], [E.sub.j]) = [[delta].sub.ij][mu][E.sub.1], (3.1)

where i, j > 1. In case that the mean curvature vector does not vanish, we see that the Lagrangian submanifold is H-umbilical if and only if we can write:

h(X, Y)= [alpha]<JX, H> <JY, H> H

+ [beta]<H,H>{<X,Y>H + <JX,H>JY + <JY,H>JX}, (3.2)

for tangent vectors X, Y, Z with

[alpha] = [lambda] - 3[mu] / [[gamma].sup.3], [beta] = [mu] / [[gamma].sup.3], [gamma] = [lambda] + (n - 1)[mu] / n.

Moreover, from [7] and [8], when n > 3, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

It is known, see [4] and [12], that [lambda] = 3[mu], characterizes the Whitney spheres (or their analogs in complex hyperbolic space) They are given by:

Example 1. Whitney sphere in [[??].sup.n] (see [1], [2], [5], [16]). It is defined as the Lagrangian immersion of the unit sphere [[??].sup.n], centered at the origin of [[??].sup.n+1], in [[??].sup.n], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Example 2. Whitney spheres in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see [4], [12]). They are a one- parameter family of Lagrangian spheres in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration.

Example 3. Whitney spheres in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see [4], [12]). They are a one- parameter family of Lagrangian spheres in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration.

Example 4. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the (n-1)-dimensional real hyperbolic space, following [4] (cf. [12]), we define a one-parameter family of Lagrangian embeddings

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration.

Example 5. Following [4] (cf. [12]), we define a one-parameter family of Lagrangian embeddings

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration.

In literature, several characterizations of the Whitney spheres exist. We recall the following, which follows by combining the Main Theorem and Lemma 3.4 of [12].

Theorem 3.1. Let [M.sup.n] be as described in (3.4)-(3.8). Then there exists a local function k on [M.sup.n] such that the covariant derivative of the second fundamental form satisfies

([nabla]h)(X, Y, Z) = [kappa](<Y, Z>JX + <X, Z>JY + <X, Y>JZ). (3.9)

Conversely any non-parallel Lagrangian submanifold satisfying the above property is congruent with a Whitney sphere.

We will now show that they always have isotropic cubic tensor. From (3.9), for any unit vector v [member of] [T.sub.p][M.sup.n] we obtain that

<([nabla]h)(v, v, v), ([nabla]h)(v, v, v)> = 9[[kappa].sup.2],

which is independent of the choice of the unit vector v. Hence [M.sup.n] has isotropic cubic tensor. This completes the proof of Theorem 1.2.

Another important class of H-umbilical Lagrangian immersions are the ones with isotropic second fundamental form. They correspond with [lambda] = -[mu]. Also in that case a complete classification has been obtained. Those which are not totally geodesic are locally described by the following examples with n [greater than or equal to] 3.

Example 6. Following [2], [13] (cf. [8]), we define a Lagrangian immersion in [[??].sup.n], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

Example 7. Following [13] (cf. [7], [18]), we define a one-parameter family of Lagrangian embeddings in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4) is the Hopf fibration and O is given by the following immersion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

where

[z.sub.1] = sinh[y(t)]exp{i[[integral].sup.t.sub.0]coth[y(s)][square root of 1 - y'[(s).sup.2]]ds} / cosh[y(t)]exp{i[[integral].sup.t.sub.0]tanh[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

[z.sub.2] = 1 / cosh[y(t)]exp{i[[integral].sup.t.sub.0]tanh[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

with y(t) determined by

-2(2[cosh.sup.2] y(t) - 3)(y'[(t).sup.2] - 1) + y"(t) sinh2y(t) = 0.

Example 8. Following [13] (cf. [7]), we define a one-parameter family of Lagrangian embeddings in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration and [PHI] is given by the following immersion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

where

[z.sub.1] = cosh[y(t)]exp{i[[integral].sup.t.sub.0]tanh[y(s)][square root of 1 - y'[(s).sup.2]]ds} / sinh[y(t)]exp{i[[integral].sup.t.sub.0]coth[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

[z.sub.2] = 1 / sinh[y(t)]exp{i[[integral].sup.t.sub.0]coth[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

with y(t) determined by

-2(1 + 2[cosh.sup.2] y(t))(y'[(t).sup.2] - 1) + y"(t) sinh 2y(t) = 0.

Example 9. Following [13] (cf. [7]), we define a one-parameter family of Lagrangian embeddings in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration and [PHI] is given by the following immersion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

where

[z.sub.1] = 1 / cos[y(t)]exp{i[[integral].sup.t.sub.0]tan[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

[z.sub.2] = sin[y(t)]exp{i[[integral].sup.t.sub.0]cot[y(s)][square root of 1 - y'[(s).sup.2]]ds} / cos[y(t)]exp{i[[integral].sup.t.sub.0]tan[y(s)][square root of 1 - y'[(s).sup.2]]ds} ,

with y(t) determined by

2(2 - cos2y(t))(y'[(t).sup.2] - 1) + y"(t) sin2y(t) = 0.

Example 10. Following [13] (cf. [7]), we define a one-parameter family of Lagrangian embeddings in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hopf fibration and [PHI] is given by the following immersion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.14)

where [mu](t) = 1 / cosh(c[+ or -]3t) and k{t) = [- or +] tanh (c [+ or -] 3t) for some constant c.

If [M.sup.n] is one of the examples 6-10, then following [7], [8], [13] and [18], we get that the second fundamental form of [M.sup.n] takes the form in (3.1) with [mu] = -[lambda], hence by (3.2) we find for arbitrary local vector fields that

h(U, V) = 4[lambda]<U, [E.sub.1]> <V, [E.sub.1]>J[E.sub.1] - [lambda]<U, V>J[E.sub.1] - [lambda]<U, [E.sub.1]>JV - [lambda]<V, [E.sub.1]>JU, (3.15)

where [lambda] is a locally defined function on [M.sup.n].

Now let p [member of] [M.sup.n]. We call [E.sub.1] ( p) = [e.sub.1] and let w, u, v [member of] [T.sub.p][M.sup.n] where w is orthogonal to [e.sub.1]. Then, using normal coordinates, we can extend u and v to local vector fields U and V defined in the neighborhood of the point p and satisfying [[nabla].sub.w]U = [[nabla].sub.w]V = 0 at point p. By (2.3) and (3.3) we calculate that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.16)

which gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.17)

By (3.15) we have h([E.sub.1], [E.sub.1]) = [lambda]J[E.sub.1], hence by (3.3) we can calculate that

([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1])= [e.sub.1]([lambda])J[e.sub.1]. (3.18)

Let v be an arbitrary unit vector in [T.sub.p][M.sup.n], if v is neither parallel nor orthogonal with [e.sub.1], then there exists a unit vector u which is orthogonal to [e.sub.1] such that v = cos [theta][e.sub.1] + sin [theta]u, hence by (3.17) and (3.18) we can calculate that

([nabla]h)( v, v, v) = [e.sub.1] ([lambda])(cos 3[theta] J[e.sub.1] - sin 3[theta] Ju). (3.19)

(3.17), (3.18) and (3.19) imply that [M.sup.n] has isotropic cubic tensor. Hence we have shown that an H-umbilical isotropic Lagrangian submanifold of a complex space form also has isotropic cubic tensor. This completes the proof of Theorem 1.3.

Remark 3.2. In [18], L. Vrancken gave a complete classification of the isotropic Lagrangian submanifolds in complex projective space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see also [10]), n [greater than or equal to] 3. Similarly the isotropic Lagrangian submanifolds in [[??].sup.n] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with n [greater than or equal to] 3, were classified by H. Li and X. Wang in [13] (see also [10]). Combining these results, we get that if [M.sup.n] is an isotropic Lagrangian submanifold of [[bar.M].sup.n] (4c) with n [greater than or equal to] 3 then [M.sup.n] is totally geodesic or has parallel second fundamental form or is H- umbilical and therefore congruent to one of the examples 6-10. In all cases it follows that [M.sup.n] also must have isotropic cubic tensor. This shows the following theorem.

Theorem 3.3. Let [M.sup.n] be an n-dimensional (n [greater than or equal to] 3) Lagrangian submanifold of a complex space form. If [M.sup.n] has isotropic second fundamental form, then [M.sup.n] also has isotropic cubic tensor.

4 Some lemmas

From now on we will always assume that [M.sup.n] is a Lagrangian submanifold of [[bar.M].sup.n] (4c) with isotropic cubic tensor, where [[bar.M].sup.n] (4c) is an n-dimensional complex space form with constant holomorphic sectional curvature 4c. We will also assume that [M.sup.n] is not a parallel Lagrangian submanifold, i.e. we will assume that Y [not equal to] 0. We have the following lemma.

Lemma 4.1. Let [M.sup.n] be a Lagrangian submanifold of [[bar.M].sup.n] (4c) with isotropic cubic tensor. Let f (v) = <[nabla]h (v, v, v), Jv> be a function on the unit tangent bundle,i.e. on [UM.sup.n.sub.p] = {v [member of] [T.sub.p][M.sup.n][parallel]|v[parallel] = 1}. Let e1 denote a vector where f attains its maximum with f ([e.sub.1]) = [a.sub.1]. Then [absolute value of [a.sub.1]] = Y( p) and for any u, a unit vector which is orthogonal to [e.sub.1], we have

(i) <([nabla]h)([e.sub.1],[e.sub.1],[e.sub.1]), Ju> = 0.

(ii) -[a.sub.1] + 3<([nabla]h)([e.sub.1], [e.sub.1], u), Ju> [less than or equal to] 0. Moreover, if the equality holds we must have <([nabla]h)(u, u, u), J[e.sub.1]> = 0.

Proof. Let

g(t) = <([nabla]h)([e.sub.1] cos t + u sin t, [e.sub.1] cos t + u sin t, [e.sub.1] cos t + u sin t), J([e.sub.1] cos t + u sin t)>.

As f attains its maximum at the vector [e.sub.1], with f ([e.sub.1]) = [a.sub.1], we see that g(t) attains its maximum value at t = 0, which implies that

g' (0) = 4<([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1]), Ju> = 0, (4.1)

g"(0) = 4(-<([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1]),J[e.sub.1]> + 3<([nabla]h)([e.sub.1], [e.sub.1], u), Ju>) [less than or equal to] 0. (4.2)

(4.1) implies (i). Moreover, if the equality holds in (4.2), we must have g"(0) = 0. Using (4.1), we obtain g'''(0) = 24<([nabla]h)(u, u, u), J[e.sub.1]>, from which (ii) follows.

From (4.1) we have that ([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1]) and J[e.sub.1] are parallel. Since f ([e.sub.1]) = [a.sub.1], which implies that ([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1]) = [a.sub.1]J[e.sub.1], we obtain that Y (p) = [a.sub.1].

We now define a linear operator A on [T.sub.p][M.sup.n] by defining

A(v) [??] - J([nabla]h)([e.sub.1], [e.sub.1], v).

Note that from (2.10) it follows that A is a symmetric operator. Also, we know that [e.sub.1] is an eigenvector of A with eigenvalue [a.sub.1]. Therefore, we can choose an orthonormal basis {[e.sub.1],...,[e.sub.n]} of [T.sub.p][M.sup.n] which diagonalizes A, i.e. A([e.sub.i]) = [a.sub.i][e.sub.i], which means ([nabla]h)([e.sub.1], [e.sub.1], [e.sub.1]) = [a.sub.i]J[e.sub.i].

Let G(t) = [parallel] ([nabla]h)([e.sub.1] cos t + [e.sub.i] sin t, [e.sub.1] cos t + [e.sub.i] sin t, [e.sub.1] cos t + [e.sub.i] sin t) [parallel][sup.2], i > 1, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

On the other hand, since [M.sup.n] has isotropic cubic tensor we have

G(t) [equivalent to] [a.sup.2.sub.1] [equivalent to] [a.sup.2.sub.1] ([cos.sup.2]t + [sin.sup.2]t)[sup.3] [equivalent to] [a.sup.2.sub.1] ([cos.sup.6]t + 3[cos.sup.4]t [sin.sup.2]t + 3[cos.sup.2]t [sin.sup.4]t + [sin.sup.6]t) (4.4)

By comparing the coefficients in (4.3) and (4.4), we get

[a.sup.2.sub.1] = 3[a.sup.2.sub.i] + 2[a.sub.1][a.sub.i], (4.5)

([a.sub.1] + 9[a.sub.i]) <([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]),J[e.sub.1]> = 0, (4.6)

[a.sup.2.sub.1] = 3[parallel]([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i])[parallel][sup.2] + 2[a.sub.i]f ([e.sub.i]), (4.7)

<([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]), ([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i])>, = 0. (4.8)

From (4.5) we have [a.sub.i] is either 1/3 [a.sub.1] or -[a.sub.1], hence by use of (4.6) we obtain that [a.sub.1] = 0 or

<([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]),J[e.sub.1]> = 0, i > 1. (4.9)

From now on, we will assume that n = 3.

Proposition 4.2. Let [M.sup.3] be a 3-dimensional Lagrangian submanifold of [[bar.M].sup.3] (4c) with isotropic cubic tensor. Let p [member of] [M.sup.3], then either

(i) [nabla]h vanishes identically at the point p, or

(ii) [nabla]h takes the following form

([nabla]h) (u, v, w) =1/3 [a.sub.1] (<v, w> Ju + <u, w> Jv + <u, v> Jw), [for all] u, v, w [member of] [T.sub.p][M.sup.3], (4.10)

or

(iii) there exists an orthonormal basis {[e.sub.1], [e.sub.2], [e.sub.3]} of [T.sub.p][M.sup.3] such that [nabla]h takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.11)

Proof. When [a.sub.1] = 0, by the choice of [e.sub.1] and the fact that <([nabla]h)(X, Y, Z), JW> is totally symmetric, we immediately get that M3 has parallel second fundamental form.

When [a.sub.1] [not equal to] 0, we have [a.sub.i] is either 1/3[a.sub.1] or - [a.sub.1], so we need to consider three cases.

Case (i): [a.sub.2] = [a.sub.3] = 1/3[a.sub.1].

Let L = span{[e.sub.2], [e.sub.3]}, by (4.9) we have

<([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]),J[e.sub.1]> = 0, i = 2,3,

so by linearization using the symmetry of [nabla]h we have

<([nabla]h)(u, v, w),J[e.sub.1]> = 0, [for all] u, v, w [member of] L,

which implies

<([nabla]h)([e.sub.1], [e.sub.i], [e.sub.i]),J[e.sub.j]> = 0, i, j = 2,3.

We also have <([nabla]h)([e.sub.1], [e.sub.i], [e.sub.i]),J[e.sub.1]> = [a.sub.i], hence we get

([nabla]h)([e.sub.1], [e.sub.i], [e.sub.i]) = 1/3[a.sub.1]J[e.sub.1], i = 2,3. (4.12)

By (4.7) and (4.12) we have

([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]) = [a.sub.1]J[e.sub.1], i = 2,3. (4.13)

Since L is the eigenspace of A with eigenvalue 1/3[a.sub.1], we have that the previous formulas are not only valid for [e.sub.2], [e.sub.3] but for any unit vector v in L. So we get

([nabla]h)([e.sub.1], v, v) = 1/3[a.sub.1][parallel]v[parallel][sup.2]J[e.sub.1], [for all] v [member of] L, (4.14)

and

([nabla]h)(v, v, v) = [a.sub.1][parallel]v[parallel][sup.2]Jv, [for all] v [member of] L (4.15)

Let v = 1/[square root of 2] ([e.sub.2] [+ or -] [e.sub.3]), we get

([nabla]h)(v,v,v) = 1 / 2[square root of 2] (([nabla]h)([e.sub.2],[e.sub.2],[e.sub.2]) [+ or -] ([nabla]h)([e.sub.3],[e.sub.3],[e.sub.3]) [+ or -] 3([nabla]h)([e.sub.2], [e.sub.2], [e.sub.3]) + 3([nabla]h)([e.sub.2], [e.sub.3], [e.sub.3])). (4.16)

(4.15) and (4.16) imply that

([nabla]h)([e.sub.j],[e.sub.j],[e.sub.k]) = 1/3[a.sub.1]J[e.sub.k], 2 [less than or equal to] j [not equal to] k [less than or equal to] 3. (4.17)

Combining all the formulas above, we get

{([nabla]h)([e.sub.i], [e.sub.i], [e.sub.i]) = [a.sub.1]J[e.sub.i], ([nabla]h)([e.sub.i], [e.sub.i], [e.sub.j]) = 1/3[a.sub.1]J[e.sub.j], ([nabla]h)([e.sub.1], [e.sub.2], [e.sub.3]) = 0, 1 [less than or equal to] i [not equal to] j[less than or equal to] 3.

(4.18)

Using (4.18), we see that for u, v, w [member of] {[e.sub.1], [e.sub.2], [e.sub.3]} we get

([nabla]h)(u,v,w) = 1/3[a.sub.1](<u, v> Jw + <v, w> Ju + <u, w> Jv) ).

As two tensors which coincide on a basis, coincide for any tangent vectors, we complete the proof in this case.

Case (ii): [a.sub.2] = 1/3[a.sub.1], [a.sub.3] = -[a.sub.1]. Since [a.sub.1] is the maximum value of f, by (4.2) we have -[a.sub.1] [less than or equal to] -3[a.sub.3] = 3[a.sub.1] , which implies that [a.sub.1] [greater than or equal to] 0. As we assume that [a.sub.1] [not equal to] 0, we get [a.sub.1] > 0. In this case, we have

{ ([nabla]h)([e.sub.1],[e.sub.1],[e.sub.2]) = 1/3[a.sub.1]J[e.sub.2], ([nabla]h)([e.sub.1],[e.sub.1],[e.sub.3]) = -[a.sub.1]J[e.sub.3],

(4.9), (4.19) and lemma 4.1 imply

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.20)

We assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.21)

by (4.7) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.22)

so we get[beta] = 0 and f ([e.sub.3]) = [a.sub.1], which means that the function also attains a maximum at [e.sub.3]. Consequently we also obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.23)

From (4.21) and (4.23) we get [e.sub.2] is an eigenvector of the operator - J([nabla]h)([e.sub.3], [e.sub.3], -), so we can assume

([nabla]h)([e.sub.3], [e.sub.3], [e.sub.2]) = bJ[e.sub.2], (4.24)

with b = 1/3[a.sub.1] or b = -[a.sub.1] since [e.sub.3] is also a maximum vector.

(4.20) and (4.23) imply that ([nabla]h) ([e.sub.2], [e.sub.2], [e.sub.2]) is parallel with J[e.sub.2], so we get

([nabla]h) ([e.sub.2], [e.sub.2], [e.sub.2]) = [+ or -][a.sub.1]J[e.sub.2]. (4.25)

(4.22) and (4.25) imply that [alpha] = 0 or [alpha] = [+ or -]2/3[a.sub.1], noting that we can always change the direction of [e.sub.3] to make [alpha] [greater than or equal to] 0 without changing the other components of [nabla]h, so we have to consider four subcases.

Case (ii-1): [alpha] = 0, b = 1/3[a.sub.1].

In this case, [e.sub.1], [e.sub.2] and [e.sub.3] are all maximum directions of f and [nabla]h takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.26)

Let y = [y.sub.1][e.sub.1] + [y.sub.2][e.sub.2] + [y.sub.3][e.sub.3], [y.sup.2.sub.1] + [y.sup.2.sub.2] + [y.sup.2.sub.3] = 1, by using (4.26) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.27)

we also have

[parallel]([nabla]h)(y, y, y)[parallel][sup.2] [equivalent to] [a.sup.2.sub.1]([y.sup.2.sub.1] + [y.sup.2.sub.2] + [y.sup.2.sub.3])[sup.3]. (4.28)

By comparing the coefficients of [y.sup.2.sub.1][y.sup.2.sub.2][y.sup.2.sub.3] in (4.27) and (4.28), we get a contradiction, so this case can't happen.

Case (ii-2): [alpha] = 0, b = -[a.sub.1].

In this case, [e.sub.1], [e.sub.2] and [e.sub.3] are all maximum directions of f and [nabla]h takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.29)

If we take [[??].sub.1] = [e.sub.3], [[??].sub.2] = [e.sub.2], [[??].sub.3] = [e.sub.1], [[??].sub.1] = [a.sub.1], then we get [nabla]h in case (ii-2) takes the same form with (4.11).

Case (ii-3): [alpha] = 2/3[a.sub.1], b = 1/3[a.sub.1].

In this case, [e.sub.1] and [e.sub.3] are both maximum directions of f, [e.sub.2] is a minimal direction of f and [nabla]h takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.30)

If we take [[??].sub.1] = ([e.sub.1] + [e.sub.3])/[square root of 2], [[??].sub.2] = [e.sub.2], [[??].sub.3] = {[e.sub.1] - [e.sub.3])/[square root of]2, [[??].sub.1] = -[a.sub.1],we obtain that [nabla]h in case (ii-3) takes the same form with (4.11).

Case (ii-4): [alpha] = 2/3[a.sub.1], b = -[a.sub.1].

In this case, we get [e.sub.2] is a minimal direction, by an analogous argument as for [e.sub.1] we can obtain that the eigenvalue for the operator - J ([nabla]h)([e.sub.2], [e.sub.2], - ) is either [a.sub.1] or - 1/3[a.sub.1].

However, using (4.19), (4.21) and (4.24), we have the following equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.31)

which implies the eigenvalue of the operator - J([nabla]h)([e.sub.2], [e.sub.2], - ) is neither [a.sub.1] nor - 1/3[a.sub.1]. So this case can't happen.

Case (iii): [a.sub.2] = [a.sub.3] = -[a.sub.1].

In this case, by a analogous argument with case (ii) we get [e.sub.1], [e.sub.2] and [e.sub.3] are all maximum directions of f and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.32)

with c = -[a.sub.1] or c = 1/3[a.sub.1].

If c = -[a.sub.1], we can easily get contradiction after a similar argument with case (ii-1). 1

If c = 1/3[a.sub.1], we deduce that [nabla]h takes the form in (4.11). This completes the proof of Proposition 4.2.

5 Proof of Theorem 1.1

By Proposition 4.2 and the assumption that there are no points for which ([nabla]h) vanishes identically, a continuity argument using [parallel]([nabla]h)[parallel] implies that we deduce that either (4.10) holds everywhere on [M.sup.3] or [nabla]h takes the form as (4.11) everywhere on [M.sup.3]. In the first case, by the results of [12], see also Theorem 3.1 we obtain that [M.sup.3] is locally isometric with a Whitney sphere (or their analogs in complex hyperbolic space).

Hence we may assume that (4.11) holds everywhere on [M.sup.3]. Let {[F.sub.1], [F.sub.2], [F.sub.3]} be an arbitrary orthonormal frame defined in a neighborhood of the point p and let {[f.sub.1], [f.sub.2], [f.sub.3]} denote the corresponding orthonormal basis at the point p. Then, if we denote by [??] the mean curvature vector, it follows that at the point p we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From this and (4.11) we see that [e.sub.1] is characterized as belonging to a 1- dimensional eigenspace of -J[[nabla].sup.[perpendicular to]][??] with eigenvalue - 1/3[a.sub.1]. We also see that the eigenspaces of this operator have constant dimensions. Hence using the classical theorem of Kobayashi and Nomizu (see page 38 of [11]), we see that [e.sub.1] can be extended to a differentiable vector field [E.sub.1] on a neighborhood of p such that at each point the function f attains a maximum at [e.sub.1]. Note that in order to have the form (4.11) on a neighborhood it is now sufficient to take local orthonormal vector fields [E.sub.2] and [E.sub.3] orthogonal to [E.sub.1]. Note that we still have some rotational freedom. Indeed rotating the vector fields [E.sub.2] and [E.sub.3] over an angle [theta] (where [theta] is a local function) preserves the expression (4.11). So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

We now write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.4)

As mentioned before, from (5.1), if we choose another basis {[[??].sub.2], [[??].sub.3]} by rotating {[E.sub.2], [E.sub.3]}, we preserve the form of [nabla]h. By making such a rotation we may always assume that [y.sub.3] = 0. Note that if moreover, [y.sub.1] = [y.sub.2] on an open set, we recover again the same rotation freedom. So in that case we may assume that [x.sub.2] = 0. From now on we will always assume that we have made the appropriate rotations. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.5)

By (2.7) we have [S.sub.ijkl] = 0. Let i = k = 1, j = l = 2 in (5.5), from (5.1)-(5.4) we get

[Z.sub.113] = -3([b.sub.4]([y.sub.1] - [y.sub.2])+[b.sub.i]([x.sub.3] - [z.sub.2])+[b.sub.2]([x.sub.2] - [z.sub.3])) / 4[a.sub.1]. (5.6)

Let i = 1, j = k = l = 2 in (5.5), we get

[z.sub.212] =3[b.sub.1][y.sub.1]-3[b.sub.4][z.sub.2]-[E.sub.1]([a.sub.1]) / 4[a.sub.1]. (5.7)

Let i = 1, j = 2, k = l = 3 in (5.5), we get

[z.sub.213] =3([b.sub.2][y.sub.2] + [b.sub.4][z.sub.3]) / 4[a.sub.1]. (5.8)

Let i = k = 1, j = l = 3 in (5.5), we get

[z.sub.112] = 3([b.sub.5]([y.sub.1]-[y.sub.2])+[b.sub.2](- [x.sub.3]+[z.sub.2])+[b.sub.3](-[x.sub.2]+[z.sub.3])) / 4[a.sub.1] (5.9)

Let i = 1, j = 3, k = l = 2 in (5.5), we get

[Z.sub.312] = 3([b.sub.2][y.sub.1]+[b.sub.5][z.sub.2]) / 4[a.sub.1]. (5.10)

Let i = 1, j = k = l = 3 in (5.5), we get

[Z.sub.313] = 3[b.sub.3][y.sub.2]-3[b.sub.5][z.sub.3]-[E.sub.1]([a.sub.1]) / 4[a.sub.1]. (5.11)

Let i = 2, j = 3, k = l = 1 in (5.5), we get

[E.sub.2]([a.sub.1]) = -[b.sub.6][x.sub.2] + [b.sub.5]([x.sub.1] - 2[y.sub.2]) + 2[b.sub.4][y.sub.3], [E.sub.3]([a.sub.1]) = -[b.sub.6][x.sub.3] + [b.sub.4]([x.sub.1] - 2[y.sub.1]) + 2[b.sub.5][y.sub.3]. (5.12)

By using (5.6)-(5.12), we obtain [S.sub.ijkl] [equivalent to] 0 is equivalent to the following 22 equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.13)

By the Gauss equation (2.4) we get moreover that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.14)

First we look at the case that [b.sup.2.sub.1] + [b.sup.2.sub.2] + [b.sup.2.sub.3] + [b.sup.2.sub.4] + [b.sup.2.sub.5] + [b.sup.2.sub.6] = 0, i.e. [b.sub.1] = [b.sub.2] = [b.sub.3] = [b.sub.4] = [b.sub.5] = [b.sub.6] = 0. Assume that this is true in a neighborhood of the point p. Hence [M.sub.3] is flat, i.e. <R([E.sub.i], [E.sub.j])[E.sub.k], [E.sub.l]> = 0, [for all]i, j, k, l = 1, 2, 3. By (2.4) we get

<h([E.sub.i], [E.sub.k]),h([E.sub.j], [E.sub.l])> - <h([E.sub.i], [E.sub.l]),h([E.sub.j], [E.sub.k])> + c(<[E.sub.i], [E.sub.k]> <[E.sub.j], [E.sub.l]> - <[E.sub.i], [E.sub.l]> <[E.sub.j], [E.sub.k]>) = 0, [for all]i, j, k, l = 1, 2, 3, (5.15)

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.16)

We insert (5.1) and (5.2) into (5.16) and obtain [x.sub.1] = [x.sub.2] = [x.sub.3] = [y.sub.1] = [y.sub.2] = [z.sub.1] = [z.sub.2] = [z.sub.3] = [z.sub.4] = 0, in a neighborhood of p. Hence [M.sup.3] is totally geodesic, and therefore also has parallel second fundamental form. This is a contradiction with [a.sub.1] [not equal to] 0.

When [b.sup.2.sub.1] + [b.sup.2.sub.2] + [b.sup.2.sub.3], + [b.sup.2.sub.4] + [b.sup.2.sub.5] + [b.sup.2.sub.6] [not equal to] 0, we can look at the system (5.13) as a system of linear equations in the variables [x.sub.1], [x.sub.2], [x.sub.3], [y.sub.1], [y.sub.2], [z.sub.1], [z.sub.2], [z.sub.3], [z.sub.4] under the condition (5.14). Note that if [x.sub.1] = [x.sub.2] = [x.sub.3] = [y.sub.1] = [y.sub.2] = [z.sub.1] = [z.sub.2] = [z.sub.3] = [z.sub.4] = 0, [M.sup.3] is totally geodesic which is a contradiction. Henceforth we are only interested in nontrivial solutions. Note also that if [y.sub.1] = [y.sub.2] (on an open set), by the choice of frame we also must have that [x.sub.2] = 0.

Using the Reduce command of Mathematica, more precisely using

Reduce[{eq2 == 0, eq3 == 0, eq6 == 0, eq7 == 0, eq19 == 0, eq22 == 0, eq20 == 0, eq21 == 0, eq9 == 0, eq10 == 0, eq13 == 0, eq14 == 0, eq15 == 0, eq18 == 0, eq17 == 0, eq16 == 0, eq8 == 0, eq11 == 0, eq12 == 0, eq4 == 0, eq5 == 0, eq1 == 0}, {[x.sub.1], [y.sub.1], [y.sub.2], [x.sub.2], [x.sub.3], [z.sub.1], [z.sub.2], [z.sub.3], [z.sub.4]}]

which is particularly adapted for solving a system of linear equations with parameters, we find that the system (5.13) only has nontrivial real solutions in the following cases:

(i) [b.sub.5] = [b.sub.4] = [b.sub.2] = 0, [b.sub.1] = [b.sub.3] = [b.sub.6] [not equal to] 0, [y.sub.1] = [y.sub.2] = - [x.sub.1], [x.sub.2] = 0, [z.sub.1] = [z.sub.3] = 0, [z.sub.2] = - 1/3[x.sub.3] and [z.sub.4] = - [x.sub.3],

(ii) [b.sub.5] = [b.sub.4] = [b.sub.3] = [b.sub.2] = [b.sub.1] = 0, [b.sub.6] [not equal to] 0, [y.sub.1] = [y.sub.2], [x.sub.2] = [x.sub.3] = 0, [z.sub.1] = [z.sub.2] = [z.sub.3] = [z.sub.4] = 0,

(iii) [b.sub.6] = [b.sub.5] = [b.sub.4] = [b.sub.2] = [b.sub.1] = 0, [b.sub.3] [not equal to] 0, [y.sub.1] = [y.sub.2] = [x.sub.1] = 0, [x.sub.2] = [x.sub.3] = 0, [z.sub.2] = [z.sub.3] = [z.sub.4] = 0,

(iv) [b.sub.6] = [b.sub.5] = [b.sub.4] = [b.sub.3] = [b.sub.2] = 0, [b.sub.1] [not equal to] 0, [y.sub.1] = [y.sub.2] = [x.sub.1] = 0, [x.sub.2] = 0, [z.sub.1] = [z.sub.3] = 0, [z.sub.2] = [x.sub.3],

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(vi) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(vii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(viii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ix) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(x) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(xi) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now look at the above solutions in more detail, taking also into account (5.14) and the fact that [M.sup.3] has isotropic cubic tensor. If necessary by restricting to an open dense subset of [M.sup.3,] we may assume that a solution remains valid on an open set. We get:

(i) [b.sub.5] = [b.sub.4] = [b.sub.2] = 0, [b.sub.1] = [b.sub.3] = [b.sub.6] [not equal to] 0. From [b.sub.1] = [b.sub.6] and (5.14), we immediately obtain that also 3[x.sup.2.sub.1] + 5[x.sup.2.sub.3]/9 = 0. This implies that all components of the second fundamental form vanish, which is a contradiction.

(ii) [b.sub.1] = [b.sub.2] = [b.sub.3] = [b.sub.4] = [b.sub.5] = 0, [b.sub.6] [not equal to] 0, in this case, using also (5.14), we get [x.sub.1] = [y.sub.1] - c/[y.sub.1], [x.sub.2] = [x.sub.3] = 0, [y.sub.2] = [y.sub.1], [z.sub.1] = [z.sub.2] = [z.sub.3] = [z.sub.4] = 0. Moreover, [b.sub.6] = -c - [y.sup.2.sub.1] [not equal to] 0. So by (5.9) and (5.11) we get [z.sub.112] = [z.sub.113] = 0. Next from (2.3), (4.11), (5.2) and (5.3) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, as [a.sub.1] [not equal to] 0, we get that c = -2[y.sup.2.sub.1]. However, this implies that [y.sub.1] is constant and hence [a.sub.1] vanishes. This is a contradiction.

(iii) [b.sub.1] = [b.sub.2] = [b.sub.4] = [b.sub.5] = [b.sub.6] = 0, [b.sub.3] [not equal to] 0, in this case a contradiction follows from (5.14).

(iv) [b.sub.2] = [b.sub.3] = [b.sub.4] = [b.sub.5] = [b.sub.6] = 0, [b.sub.1] [not equal to] 0, in this case we get [x.sub.1] = [x.sub.2] = [y.sub.1] = [y.sub.2] = 0, [z.sub.1] = 0, [z.sub.2] = [x.sub.3], [z.sub.3] = 0, [z.sub.4] = [x.sub.3] - c/[x.sub.3], [b.sub.1] = -c - [x.sup.2.sub.3] [not equal to] 0. We proceed again as for case (ii) in order to obtain a contradiction.

(vii) [b.sub.4] = [b.sub.5] = [b.sub.6] = 0, [b.sub.1][b.sub.3] = [b.sup.2.sub.2] [not equal to] 0, [x.sub.1] = [y.sub.1] = [y.sub.2] = [x.sub.2] = [x.sub.3] = 0, [z.sub.2] = - [b.sub.1]/[b.sub.2] [z.sub.1],[z.sub.2] = [b.sup.2.sub.1]/[b.sup.2.sub.2] [z.sub.1],[z.sub.2] = - [b.sup.3.sub.1]/[b.sup.3.sub.2] [z.sub.1]. Using (5.14) we find that [b.sub.2] = 0 which leads to a contradiction.

(viii) and (ix) In both cases we have [b.sub.4] = [b.sub.2] = [b.sub.1] = 0, [b.sup.2.sub.5] = [b.sub.3][b.sub.6] [not equal to] 0 and the solution can be rewritten as [x.sub.1] = [b.sub.5]/[b.sub.3]([x.sub.2] + 2[z.sub.3]), [x.sub.3] = 0, [y.sub.1] = [b.sub.5]/[b.sub.3][z.sub.3] + [b.sub.5]/[b.sub.3]([x.sub.2] - [z.sub.3]), [y.sub.2] = [b.sub.5]/[b.sub.3][z.sub.3], [y.sub.3] = 0, [z.sub.i] = [b.sup.2.sub.3]/[b.sup.2.sub.5]([x.sub.2] - [z.sub.3]) + 3[z.sub.3], [z.sub.2] = 0, [z.sub.4] = 0. After a direct calculation, we get [z.sub.112] = [z.sub.113] = [z.sub.213] = [z.sub.312] = 0, [z.sub.212] = [z.sub.313] = - [E.sub.1]([a.sub.1])/4[a.sub.1] Hence by the definition of the curvature <R([E.sub.i], [E.sub.j])[E.sub.k], [E.sub.l]>, we can calculate that

<R([E.sub.2], [E.sub.1])[E.sub.2], [E.sub.1]> = <R([E.sub.3], [E.sub.1])[E.sub.3], [E.sub.1]> = 5([E.sub.1]([a.sub.1]))[sup.2]-4[a.sub.1][E.sub.1]([E.sub.1]([a.sub.1])) / 16[a.sup.2.sub.1]

which is a contradiction with [b.sub.1] = 0, [b.sub.3] [not equal to] 0.

(v) [b.sub.5] = [b.sub.3] = [b.sub.2] = 0, [b.sub.1][b.sub.6] = [b.sup.2.sub.4], [b.sub.6] [not equal to] 0 [not equal to] [b.sub.4], [y.sub.2] = [b.sup.2.sub.1][x.sub.1]- 3[b.sup.2.sub.1][y.sub.1]+[b.sup.2.sub.4][y.sub.1]/[b.sup.2.sub.4], [x.sub.2] = 0, [x.sub.3] = [b.sub.1][x.sub.1]-2[b.sub.1][y.sub.1]/[b.sub.4], [z.sub.1] = [z.sub.3] = 0, [z.sub.2] = [b.sub.1][x.sub.3]+[b.sub.4][y.sub.1]-[b.sub.4][y.sub.2]/[b.sub.1] and [z.sub.4] = 2[b.sub.1][y.sub.1]+[b.sub.1][y.sub.2]/[b.sub.4]. Changing the roles of [E.sub.2] and [E.sub.3] this case reduces to the previous one and hence a contradiction follows in the same way.

(vi) and (x) In both cases, we get [x.sub.2] = [x.sub.3] = 0, [y.sub.1] = [y.sub.2] = -[x.sub.1], [y.sub.3] = 0, [z.sub.1] = [z.sub.2] = [z.sub.3] = [z.sub.4] = 0 and [b.sub.2] = [b.sub.4] = [b.sub.5] = 0, [b.sub.1] = [b.sub.3] = - c + 2[x.sup.2.sub.1] [not equal to] 0, [b.sub.6] = - c - [x.sup.2.sub.1], and the second fundamental form of [M.sup.3] takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So by (5.9) and (5.11) we get [z.sub.112] = [z.sub.113] = 0. Next from (2.3), (4.11), (5.2) and (5.3) we get

[nabla]h([E.sub.1], [E.sub.1], [E.sub.1]) = [E.sub.1]([x.sub.1])J[E.sub.1] = [a.sub.1]J[E.sub.1].

In case (vi), from [b.sub.6] = -[b.sub.1] we get [x.sup.2.sub.1] = -2c hence [E.sub.1] ([x.sub.1]) = 0, which together with the previous formula imply [a.sub.1] vanishes. So we get a contradiction.

In case (x), following [7], [8] (cf. [13], [18]), we obtain that [M.sup.3] is isotropic and H-umbilical and therefore locally congruent to one of the examples 6-10 with n = 3.

(xi) [b.sub.2] = [b.sub.4] = [b.sub.5] = [b.sub.6] = 0, [b.sub.3] = 2[b.sub.1] [not equal to] 0. From (5.14) it follows that c = 0 and [b.sub.3] = 2[x.sup.2.sub.3]. From (5.6) - (5.11) we get [z.sub.113] = - 3[x.sup.3.sub.3]/2[a.sub.1], [z.sub.112] = [z.sub.312] = [z.sub.213] = 0, [z.sub.212] = [z.sub.313] = - [E.sub.1]([a.sub.1])/4[a.sub.1]. It follows from (2.3), (4.11), (5.2) and (5.3) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so we get [a.sub.1] = 0, which is a contradiction.

Hence, we have completed the proof of Theorem 1.1.

Acknowledgements: The authors would like to express their thanks to the referee for some useful comments.

Received by the editors July 2010. Communicated by L. Vanhecke.

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[2] I. Castro, Lagrangian spheres in the complex Euclidean space satisfying a geometric equality. Geometriae Dedicata 70 (1998), 197-208.

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[6] B. Y. Chen, Riemannian Geometry of Lagrangian Submanifolds. Taiwanese Journal of Mathematics Vol.5. No.4. (2001), 681-723.

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[11] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. II, Interscience Publishers, New York 1963.

[12] H. Li and L. Vrancken, A basic inequality and new characterization of Whitney spheres in a complex space form. Israel Journal of Mathematics 146 (2005), 223242.

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[16] A. Ros and F. Urbano, Lagrangian submanifolds of Cn with conformai Maslov form and the Whitney sphere. J. Math. Soc. Japan 50 (1998), 203-226.

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Xianfeng Wang *,[dagger]

School of Mathematical Sciences, Nankai University, Tianjin 300071, Peoples Republic of China. E-mail: xf-wang06@mails.tsinghua.edu.cn

Haizhong Li *,[double dagger]

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China E-mail: hli@math.tsinghua.edu.cn

Luc Vrancken *

Univ. Lille Nord de France, F-59000 Lille, France; U[NABLA]HC, LAMAV, F-59313 Valenciennes, France; Katholieke Universiteit Leuven, Departement Wiskunde, BE-3001 Leuven, Belgium E-mail: luc.vrancken@univ-valenciennes.fr

* Supported by Tsinghua University-K.U.Leuven Bilateral scientific cooperation Fund.

([dagger]) Supported by NSFC grant No. 10701007

([double dagger]) Supported by NSFC grant No. 10971110
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Author:Wang, Xianfeng; Li, Haizhong; Vrancken, Luc
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Aug 1, 2011
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